Question

you pick a card at random, put it back, and then pick another card at random. what is the probability of picking an even number and then picking a number greater than 7? simplify your answer and write it as a fraction or whole number.

Explanation:

Step1: Calculate the probability of picking an even number

There are 3 cards in total (\(7\), \(8\), \(9\)). The even - numbered card is \(8\). So the probability of picking an even number \(P(\text{even})=\frac{1}{3}\) (using the formula \(P=\frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}\)).

Step2: Calculate the probability of picking a number greater than \(7\)

The numbers greater than \(7\) are \(8\) and \(9\). So the probability of picking a number greater than \(7\), \(P(\text{> 7})=\frac{2}{3}\) (since there are 2 favorable outcomes out of 3 total outcomes).

Step3: Use the multiplication rule for independent events

Since the first card is put back (independent events), the probability of both events occurring is \(P = P(\text{even})\times P(\text{> 7})\).
Substitute the values: \(P=\frac{1}{3}\times\frac{2}{3}=\frac{2}{9}\)

Answer:

\(\frac{2}{9}\)